but what is $^code?

It's a positional parameter of block that is passed to map.

The block could also have been written as { $_(|$c) } or -> $code { $code(|$c) }

What is a reasonable value for the $c?

To clarify, this method is called when you try to invoke a Junction like a subroutine or codeblock. Which of course only makes sense if the Junction contains callable eigenstates. $c holds the parameter list that was passsed to this call. IT can be empty, if you like that:

$ perl6 -e 'any({say 42}, {say 23}).()'
42
23
# with arguments:
$ perl6 -e 'any({say $^x }, { say 40 + $^x }).(2)'
2
42
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My hope is that there is a way to alter a Junction object. In Sudoku, every cell begins with the potential to be any digit

You are trying to abuse junctions as sets, which is explicitly discouraged.

I know the excitement when learning about junctions, but after experimenting a bit them I'm firmly convinced that using them for everything but simple matchers is a broad over-use, and makes your program hard to write, read and understand.